Sieve-based Empirical Likelihood under Semiparametric Conditional Moment Restrictions

In this paper we propose a new Sieve-based Locally Weighted Conditional Empirical Likelihood (SLWCEL) estimator for models of conditional moment restrictions containing …nite dimensional unknown parameters and in…nite dimensional unknown functions h. The SLWCEL is a one-step information-theoretic alternative to the Sieve Minimum Distance estimator analyzed by Ai and Chen (2003). We approximate h with a sieve and estimate both and h simultaneously conditional on exogenous regressors. Thus, the estimator permits dependence of h on endogenous regressors and . We establish consistency and convergence rates for the estimator and asymptotic normality for its parametric component of . The SLWCEL generalizes in two ways the Conditional Empirical Likelihood (CEL) of Kitamura, Tripathi and Ahn (2004). First, we construct the CEL’s dual global MD-objective function with a new weighting scheme that adapts to local inhomogeneities in the data. Second, we extend the resulting new estimator into the semiparametric environment de…ned by the presence of h. We show that the corresponding estimator of exhibits better …nite-sample properties than found in the previous literature. Keywords: Semi-/nonparametric conditional moment restrictions, empirical likelihood, sieve estimation, endogeneity. JEL Classi…cation: C13, C14, C20, C30. I am grateful to Mehmet Caner, Xiaohong Chen, George-Levi Gayle, Soiliou Daw Namoro, Taisuke Otsu, Eric Renault and Nese Yildiz for insightful comments and suggestions. I would also like to thank participants of the 16th Annual Meeting of the Midwest Econometrics Group, Cincinnati, OH, October 2006, the 2nd PhD Presentation Meeting at LSE, London, UK, January 2007, the North American Summer Meetings of the Econometrics Society, Durham, June 2007, the CESG meetings, Montreal, September 2007, and seminar participants at Georgetown, Pittsburgh, Purdue, Oxford (Nu¢ eld), Simon Fraser, Tilburg, Toronto, UNC Chapel Hill and Warwick. Address for correspondence: Martin Burda, Department of Economics, University of Toronto, Sidney Smith Hall, room 5025A, 100 St. George St., Toronto, ON, M5S 3G7, Canada. Phone: 416-978-4479. E-mail: [email protected] 1 Introduction Moment restrictions frequently provide the basis for estimation and inference in economic problems. A general framework for analyzing economic data (Y;X) is to postulate conditional moment restrictions of the form E [g (Z; 0) jX] = 0 (1) where Z (Y 0; X 0 z); Y is a vector of endogenous variables, X is a vector of conditioning variables (instruments), Xz is a subset of X; g( ) is a vector of functions known up a parameter ; and FY jX is assumed unknown. The parameters of interest 0 ( 0; h0) contain a vector of …nite dimensional unknown parameters 0 and a vector of in…nite dimensional unknown functions h0( ) (h01( ); :::; h0q( ))0: The inclusion of h0 renders the condition (1) semiparametric, encompassing many important economic models. It includes for example the partially linear regression g (Z; 0) = Y X 0 1 0 h0(X2) analyzed by Robinson (1988) and the index regression g (Z; 0) = Y h0(X 0 0) studied by Powell et al. (1989) and Ichimura (1993). Recently, Kitamura, Tripathi and Ahn (2004) analyzed the Conditional Empirical Likelihood (CEL)1 based on a parametric counterpart of (1) (with only) that was shown to exhibit …nitesample properties superior to the Generalized Method of Moments. In this paper we …rst suggest a new Locally Weighted CEL (LWCEL) that fundamentally changes the form of CEL and further improves on it in terms of …nite-sample properties. Then we extend the LWCEL to the semiparametric environment of model (1) proposing new Sieve-based Locally Weighted Conditional Empirical Likelihood (SLWCEL) estimator. The SLWCEL can be viewed as a one-step information-theoretic alternative to the Sieve Minimum Distance (SMD) estimator analyzed by Ai and Chen (2003). In the remainder of the introduction we will elaborate on the heuristic origins of both estimators, and further analysis will follow thereafter. 1.1 Conditional Moments Based on 0 Without the unknown functions h0; model (1) becomes the parametric model of conditional moment restrictions E [g (Z; 0) jX] = 0 (2) Typically, faced with the model (2) for estimation of 0; researchers would pick an arbitrary matrixvalued function a(X) and estimate the unconditional moment model E [a(X)g (Z; 0)] = 0 implied by (2) with an estimator such as the Generalized Method of Moments (GMM) (see e.g. Kitamura, 2006, 1A note on terminology: CEL is called “smoothed”and “sieve”empirical likelihood in KTA and Zhang and Gijbels (2003), respectively. Other types of smoothing have been introduced by Otsu (2003a) on moment restrictions in the quantile regression setting and hence KTA’s original method is referred to as "conditional" empirical likelihood to avoid confusion. The CEL terminology was also adopted in Kitamura (2006).